Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{a^2 + 5a}{a^2 - a - 30}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 + 5a}{a^2 - a - 30} = \dfrac{(a)(a + 5)}{(a - 6)(a + 5)} $ Notice that the term $(a + 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 5)$ gives: $q = \dfrac{a}{a - 6}$ Since we divided by $(a + 5)$, $a \neq -5$. $q = \dfrac{a}{a - 6}; \space a \neq -5$